These aspects may be useful to improve the calculation of physical quantities evaluated through the RG equations. For example, let us consider the ratio \(R\)  which is related to the total cross section for the \(e^+e^-\)  annihilation into hadrons.
This is a function of the center of mass energy squared \(s\)  and of the renormalized coupling constant \(\alpha \) . At high energies, this ratio may be written in terms of a perturbative series as [1]}, [2]}
\(R(s/\mu ^2,\alpha (\mu ^2/\Lambda ^2)) =\sum _i Q_i^2\left[1+t_1(s/\mu ^2)\alpha (\mu ^2/\Lambda ^2)+t_2(s/\mu ^2)\alpha ^2(\mu ^2/\Lambda ^2)+\cdots \right],\) 
